A noncooperative justification for egalitarian surplus sharing

Mathematical

Social Sciences

245

17 (1989) 245-261

North-Holland

A NONCOOPERATIVE EGALITARIAN Youngsub

of Economics,

Communicated Revised

FOR

SHARING

CHUN*

Department

Received

JUSTIFICATION

SURPLUS

Southern

Illinois

University,

Carbondale,

IL 62901-4515,

by H. Moulin

19 September 1 August

We consider

1987

1988

a surplus

sharing

as to how such problems

problem

faced by a group

should be solved in general.

of agents who have different

We propose

a procedure

flict and show that each agent receives an equal share of the surplus equilibria

of the game induced

Key words:

U.S.A.

Surplus

sharing

opinions

to resolve the con-

in all noncooperative

Nash

by the procedure.

problem;

noncooperative

justification;

egalitarian

surplus

sharing.

1. Introduction A fixed, finite number of agents generate a positive surplus by cooperating. Given their claims (which are the amounts invested by agents), how should the surplus be divided among them? Recently, several axiomatic justifications for particular surplus sharing solutions have been established (Aumann and Maschler, 198.5; Banker, 1981; Chun, 1988; Curiel, Maschler and Tijs, 1987; Moulin, 1985, 1987b; O’Neill, 1982; and Young, 1987, 1988). The egalitarian and proportional surplus sharing solutions play an important role in these papers. Here we consider the same problem but we take a different, noncooperative, approach. We propose that surpluses be divided on the basis of the different opinions that agents may have as to how such problems should be solved in general. In a first stage, we allow each agent to propose a surplus sharing solution satisfying three natural conditions.’ The class of solutions satisfying these conditions is quite large, and includes most of the solutions axiomatically characterized in the papers quoted above. In the second stage, the share desired by each agent is calculated using his choice of a solution. If the desired shares are feasible, i.e., if their sum is less than the sur-

* This paper is based on Chapter to Professor

William Thomson

5 of my PhD dissertation

for his helpful

advice.

at the University

1 am also grateful

Marcus Berliant for valuable comments. I retain, however, full responsibility 1Formal definitions of these conditions are given in Section 2.

0165-4896/89/$3.50

0

1989, Elsevier

Science

Publishers

of Rochester.

to Professors

B.V. (North-Holland)

I am grateful

Herve Moulin and

for any shortcomings.

Y. Chun / Egalitarian surplus sharing

246

plus, then each agent receives the amount that he asks for. Otherwise, the same solutions are applied to the revised problem obtained from the original one by setting the claim of each agent equal to the maximum of the shares assigned to him by all the proposed solutions. This procedure is repeated until the revised shares are feasible. At the final step, the limits of the desired shares are calculated. If the limits exist and their sum is less than the surplus, then each agent receives the amount specified by the limit. Otherwise, he receives nothing. We analyze this procedure as a game in which strategies are solutions satisfying the three conditions. Our main result is that each agent receives an equal share of the surplus in all noncooperative Nash equilibria of the game induced by this procedure. If an agent announces the egalitarian surplus sharing solution, then the desired shares of the other agents converge to the equal shares of the surplus. Moreover, announcing the egalitarian surplus sharing solution is a dominant strategy for the agent with the smallest claim. The approach we adopt was developed, in bargaining theory, by van Damme (1986) and used by him to give a noncooperative justification for the Nash (1950) bargaining solution in the 2-person case. Chun (1984) proposed a variant of van Damme’s procedure that gives a noncooperative justification for the KalaiSmorodinsky (1975) bargaining solution in the 2-person case. The procedure presented here can be regarded as a modification of van Damme’s procedure so that it be suitable for surplus sharing problems. However, we note that our procedure can be used to resolve conflicts among an arbitrary finite number of agents, while van Damme established his result only for the case of 2-person and its extension to the n-person case still is an open question. The paper is organized as follows. In Section 2, we describe the problem, state the axioms and define the concept of surplus sharing solutions. In Section 3, we present our main result. In Section 4, we investigate the extent to which our result is preserved under variations in our assumptions and axioms. Finally, in Section 5, we discuss

how our results

can be carried

2. Surplus sharing problem;

axioms;

over to deficit

sharing

problems.

solutions

Let N= (1, . . . . n} be the set of agents, with generic element i. A surplus sharing problem, or simply a problem, is a pair (c,s) E 81 x %++, where c=(c;)~~~, satisfying CjEN c, > 0, is the vector of claims and s> 0 is the surplus, with the following intended interpretation: each agent i, for ieN, has invested c, in some joint project, which results in a surplus s - the amount of revenue remaining after paying back the agents for the investment. How should the surplus be divided among the agents?2 We denote by .?Z3the class of all such problems.

2 For other

interpretations

of this problem,

we refer to Moulin

(1987b, 1988).

Y. Chun / Egalitarian

A deficit sharing problem3

Remark.

surplus

247

sharing

is a pair (c,s)E%:

X%++ such that

0~s~

we denote by s a surplus, or a net c ,E,,, ci. With a slight abuse of notation, revenue, for surplus sharing problems and a total revenue for deficit sharing problems. From the mathematical viewpoint, the only difference between these two of the condition, however, problems is the condition ss CieN ci. The imposition would not affect our conclusions. A surplus sharing solution, or simply a solution, is a function F: .B’ + !I??”which associates with every problem (c,s) E .!!%Ia vector F(c,s)= (F,(c,s));., such that CiEN F,(c,s)~s.

Notation.

We define

Let F be a solution. ditions:

CNE

CiGN

C;FN(C,S)G

We are interested

Efficiency (Ef ). For all problems

CiEN

c.(C,S)

in solutions

and

satisfying

so on.

the following

con-

(c, s) E 33, FN (c, s) = s.

Fair Ranking (FR). For all i, je N and for all problems

(c,s) E 33, if C; L cj, then

F;(c,s)rF,(c,s).

Progressivity (Pr). For all i, jE N and for all problems

(c,s) E 33, if C; 2 cj, then

C,F;(C,S)lC;Fj(C,S).4

Efficiency requires that the whole surplus be distributed to the agents. Fair Ranking requires that an agent with a larger claim receive a greater share of the surplus than an agent with a smaller claim. Progressivity requires that an agent with a smaller claim receive a more than proportional share of the surplus. Fair Ranking and Progressivity were first used in axiomatic studies by Moulin (1985,1987a) and Young (1987,1988) respectively. We denote by @ the class of all solutions satisfying duce the egalitarian and proportional solutions. Definition.

The egalitarian solution E is defined

for all i E N and for all problems

if c,=cj,

then F,(c,s)=F,(c,s).

by

(c, s) E B.

3 It will be discussed more in Section 5. 4 We note that Fair Ranking, or Progressivity, (c,s)~.B,

Ef, FR and Pr. Next we intro-

implies

that

for all i, j E N and

for all problems

248

Definition.

Y. Chun / Egalitarian surplus sharing

The proportional

solution P is defined

by

P;(c,s)=;s for all i E N and for all problems Note that we assumed

(c, S) E ~5’.

that c,,,>O. This is necessary

for the proportional

solution

to be well-defined. It can easily be shown that the egalitarian and proportional tions satisfy the three axioms, and therefore, belong to the family 9:.

solu-

3. Main result First, we describe our procedure. Let (c, S) E ~5’ be given. In the first stage, each agent proposes a solution in @‘, that is, a solution satisfying Ef, FR and Pr. Let F’, for i=l , . . . , n, be the choice of agent i and c’ = c. Then the following calculations are performed.

Step t. (t= 1, . ..). The desired shares of each agent, e!(c’,s) culated. If (ei(c’,s)),.,,, procedure terminates.

is feasible, Otherwise,

for i,j= 1, . . . , n, are cal-

then each agent receives what he asks for and the we go to step t + 1 with ci+l =maxjEN
Step 03. The limits of the desired shares, lim,,,

F;(c’, s) for i = 1, . . . , n, are calcu-

lated. If the limits exist and their sum is less than S, then each agent receives share specified by the limit. Otherwise, he receives nothing.

the

In our procedure, the revised claim of an agent is set equal to the maximum of all desired shares, including his own. If we have only two agents, for F’ and F2 satisfying Ef, F: (c’, s) + Ft (c’, s) 2s implies that c/(c’, S) L FL!(ct,s) for all i # j. So we can obtain the same result by setting cf+’ =Fi’(c’,s) for all t = 1, . . . and for all i = 1,2. Alternative processes of revising claims will be discussed in the next section. To facilitate the understanding of our procedure, we compare it with that suggested by van Damme (1986). First, we briefly present the latter.5 Consider a twoperson bargaining game (S’, d) where S’ c 9?>2’,is the feasible set satisfying certain restrictions and YES’ is the disagreement point. If the agents unanimously agree on a point x of S’, they obtain x. Otherwise, they obtain d. A bargaining solution is a function which associates with every bargaining game (S’, d) a point F(S’, d) E S’. Now suppose that agents have different opinions as to how such bargaining games should be solved in general. We are interested here in finding a procedure which can be used to resolve such conflicts. For this, in van Damme (1986), first each agent 5 For a detailed and rigorous presentation, see van Damme axiomatic bargaining theory, see Thomson (1988).

(1986). Also for an excellent

survey

of

Y. Chun / Egalitarian

surplus

sharing

249

U 2

I

N(S’,d) = (1,1)

d=O

a (S’,d)

(2,l) = a (S’,d)

1 Fig. 1.

i, i = 1,2, is required to announce a bargaining solution F’ satisfying a certain list of axioms. At step t, where t = 1, . . . , if (F: (S’, d), F-f@‘, d)) is feasible, then each agent receives the amount he asks for. Otherwise, we proceed to the next step with S ‘+’ = {xES’

Ix,5F’(S’,d)

for all i}.

At the final step, if lim,,, F’(S’,d)=lim,,, F*(S’,d), then agents receive the amount specified by the limit. Otherwise, they obtain d. Now, as an example, let S’ be the convex hull of (O,O), (l,O), (1,l) and (0,2), d=O, and suppose that agent 1 announces the Kalai-Smorodinsky solution6 and agent 2 announces the Nash solution. Figure 1 shows how the revisions are carried out and how the conflict is resolved.’ Now we go back to our procedure. As an example, let c = (2, l), s = 2, and suppose that agent 1 announces the proportional solution and agent 2 announces the egalitarian solution. Figure 2 shows how the revisions are carried out and how the conflict is resolved. A comparison of Figs. 1 and 2 shows how closely our procedure, for n = 2, is related to van Damme’s. However, in the case of an arbitrary finite number of agents, we take the concession of each agent to be minimal by selecting as the new claim the maximum of all previous offers including his own.

6 Given a two-person bargaining game (S, d), its Kalai-Smorodinsky (1975) solution outcome K(S, d) is the maximal point of S on the segment connecting d to a(S,d), where for each i, ai = max{x, / XE S,xz d}: its Nash (1950) solution outcome N(S, d) is the point where the product HieN (x,-d;) is maximized for XES with xrd. ’ Figure

1 is a reproduction

of Fig. 1 in van Damme

(1986).

250

Y. Chun / Egalitarian

0

surplus

sharing

1

2

Fig. 2

In the following, we show that if the agent with the smallest claim announces the egalitarian solution, then the desired shares of the other agents converge to the equal shares of the surplus. Lemma 1. Let(c,s)EB withc,?c,>...zc,begiven. Fin@, if F”=E, then lim,,, F;(c,s)=s/n.

Foralli=l,...,nandforall

Proof. First note that, since F” = E, cI= maxjE,,, F;j(c’-‘, s)rs/n for all f 2 2 and for all i. Also note that, since all F”s satisfy FR, we have c{ 2 C~Z -.. 2 CAfor all t, and therefore, Ff’(c’, S) zs/n and F,‘(c’, s) 5:/n for all t and for all i. Pr applied to the desired shares at step 2 implies that

cJ’F;(c*,s) 5 cf F,j(c’, s) Since c,‘=s/n,

inequality

(1) applied

t Fi’(c2,s) 5 c:F~(c*,s)

for all i,j. to j= n gives for all i,

which by Ef gives

or equivalently, n-1

F,‘(c2,s) + c: c Fj’(c2,s) 5 SC;. j=2

(1)

Y. Chun

Together

with inequality s

/ Egalitarian

(1) applied

n

251

to j = 2, . . . , n - 1, we have

cf F[(c2,s) + F[(c2,s) c c;

+

sharing

n-l

( > -

surplus

5

SC;.

j=2

Since

we have

(~s+c:)F;(c2,s)isc:, or equivalently, 2 F,‘(c2,s)

5

sc’

.

n-1 ps+c: n Since this inequality

This agrument, Cl

I+1

for all i, we have

holds

to all t, for t =2, . ..‘. yields

generalized

SC; 5

Now consider

n-l ps+c; n

(2)

’

the function

f(x)=R_T

f: %+ + !I?+ defined

.

-s+x n

Note that

n-l f”(x)=

-2(;;xY

2 50,

by

252

Y. Chun

/ Egalitarian

surplus

sharing

f(O)= 0, f

0 t

=f,

and lim,,, f(x) =s. The graph off is represented in Fig. 3. Let {e{) be the sequence of claims satisfying inequality (2) as an equality with ~7;= c! and E: = cf. Then we have t=2,...,

which is shown in Fig. 3. Now we show that cf
that,

and

by induction,

c;n+‘sf(c;“)

Therefore,

for all t. By definition

of Ci and S:, we know that

ct
holds for all t = 1,. . . , m. Then

(by (V),

5 f(Cr)

(since f is monotonic

-m+’ 5 c,

(by definition

of ??+I).

we have cf 5 Cf

for

for all t.

Fig. 3.

increasing),

we have

Y. Chun / Egalitarian

surplus

253

sharing

Since we know that -s 5 lim ci 5 lim E: = s

n

1--C=

n’

t+m

we have limci=?. fv+rn This implies

that,

n for all j,

lim F/(c’,s) I-co Finally,

= A.

n

by FR, we have lim F:(c’,s) = f t-011

for all i.

0

Lemma 2. Let (c, s) E 33 with cl 2 c2 2 ... L c, be given. Then to announce the egali-

tarian solution is a dominant strategy for agent n. From the proof of Lemma 1 we know that agent n cannot receive more than if he announces the egalitarian solution, then he always receives s/n as shown in Lemma 1. 0 Proof.

s/n. However,

From

Lemmas

1 and 2, we derive the following

result.

Theorem 1. Each agent receives an equal share of the surplus in all noncooperative Nash equilibria of the game induced by our procedure. Remark. In our procedure, agents are not allowed to change their choice of solutions at each step. However, as in van Damme (1986), it can be easily seen that all the results still hold if agents are allowed such changes.

4. Role of assumptions We investigate in our assumptions

and axioms

here the extent to which our results are preserved and axioms.’

under variations

4.1. Process of revising claims In section 3, the revised claim of agent i was taken to be maximum of the previous desired shares including agent i’s own claim. Here we discuss alternative ways of revising claims. 8 I am grateful

to Professor

Hew?

Moulin

for suggesting

the analysis

done in Sections

4 and 5.

Y. Chun

254

/ Egalitarian

surplus

sharing

(i) Suppose that we set ~:+t=max~,,,,~,, Z$!(c’,s) for all t=l,... and for all i=l , . . . , n. In this formulation, the revised claim of agent i is entirely determined by the other agents. Now suppose that N= { 1,2,3}, c = (300,200,lOO) and s = 100. Also assume that agents 1 and 2 announce the proportional solution and agent 3 announces the egalitarian solution. Then’we have

Fl(2,s) = F2(&) = (50, y,

T)

and

Since (F,‘(c’,s),F~(c’,s),~~(c’,~)) = (50,7, F) is feasible, we go to the next step with c2 = (50, y, F) and the conflict is never resolved. To extend the negative result for problems with n > 3, we introduce n - 3 additional agents identical to agent 2 and set s = lOOn/3. A similar argument shows that the conflict is never resolved. Given this negative result for all n 2 3, we investigate whether this alternative way of revising claims could resolve conflicts if there are only two agents. Assume that c = (25,O) and s = 10. Also suppose that agent 1 announces the proportional solution and agent 2 announces the egalitarian solution. Since F’(c’, s) = (10,O) and F2(c1,s)=(5,5),(F,i(c1,s),F~(c’,s))=(10,5) is not feasible. So we go to the next step with c2 = (5,O) and here too the conflict is never resolved. However, if we impose the additional assumption on our domain of problems that ci > 0 for all i, then the new formulation can resolve any conflicts between two agents. This additional assumption requires that agents who have made no investment should not be included in the division of surplus. We omit the proof of this result, which is similar to that of Theorem 1. and for all (ii) Suppose now that we set ci‘+lzmin. ,E,,, Fi(c’,s) for all t=l,... i=l , . . . , n. Under this formulation, the revised claim of agent i is taken to be the minimum of all previous desired shares including his own. We can easily see that we cannot impose the domain condition C~LS if we use this formulation. HOWever, if we assume that c, >O, this formulation yields the same conclusion as ours. Again, since the proof is similar to ours, we omit it. 4.2.

Axioms

Since Ef is standard

and innocuous,

we discuss

only FR and Pr.

(i) Fair Ranking; In the proof of the theorem, we apply FR only to agents with the largest claim and the smallest claim. Therefore, FR can be weakened to the following.

Weak Fair Ranking. For all i E N and for all (c, s) E 9C?,if ci 2 cj for all j E N, then

Y. Chun

/ Egalitarian

surplus

255

sharing

F, (c, s) zFJ (c, S) for all j E N and if ci 5 Cj for all j E N, then F, (c, S) i FJ (c, S) for all jeN. However,

FR cannot

Let N= (1,2,3]

be completely

and F’ be defined

F’(c, s) =

dropped

as shown in the following

(-20,20,30)

if (c,s) = (30,20,10,30),

(30,20, -20)

if (c, s) = (10,20,30,30),

1 E(c, s)

example.

by9

otherwise,

for all (c, S) E ~6’. It can easily be checked that this solution satisfies Ef and Pr, but not FR. Now suppose that c= (30,20, lo), s = 30, F’ = F3 = E and F* = F’. Then we have c*‘-‘=(30,20,10) ( C 2r= (10,20,30)

for all r=l,2

,...,

for all r= 1,2, . . . ,

and the conflict is never resolved. To extend the negative result for problems with n > 3, we introduce tional agents identical to agent 2, set s= 10n and define F’ by (lo-10420,...,

F’(c,s) =

20,30)

(30,20, . . . , 20,10-10n) r E(G s)

n - 3 addi-

if (c,s) = (30,20, . . . ,20,10, lOn), if (c,s)=(10,20

,..., 20,30,10n),

otherwise.

A similar argument shows that the conflict is never resolved. Given the negative result for all n > 3, we now investigate whether we can drop FR for the case n = 2. Although announcing the egalitarian solution is not a dominant strategy for an agent with the smallest claim any more, we can still prove the following lemma. Let $’ be the class of solutions satisfying Ef and Pr. Lemma 3. Let (c,s) E .%’ with n =2 be given. For all i, j = 1,2, i#j, F’ E @I, if FJ = E, then agent j always receives s/2.

and for all

Since the proof of this lemma is similar to that of Lemma 1, we omit it. Lemma 3 implies that, if an agent announces the egalitarian solution, he always receives s/2 and the other agent receives at most s/2. Therefore, in any noncooperative Nash equilibrium of the game induced by our procedure, both agents receive s/2. (ii) Progressivity: axiom.

First,

we discuss an alternative

formulation

9 To generate following counter-examples, we need to specify fore, F’ can easily be modified to be continuous.

F’(c,s)

of the Progressivity

only for two problems.

There-

256

Y. Chun

/ Egalitarian

surplus

sharing

Progressivity 2 (Pr2).” For all i,j E N and for all problems Cl - C’ 2

(c, s) E 3,

if c; 2 Cj, then

F; (~3S) - Fj (CTS) .

Pr2 requires that a solution should yield a smaller difference in distributed surplus than the difference in claims. Pr2 and FR’together are equivalent to order-preserveness introduced

by Aumann

and Maschler

(1985). The family of solutions

satisfying

Pr2 is fairly large, including the egalitarian solution but not the proportional solution. However, if we restrict our domain to c,zs, then the proportional solution satisfies Pr2 on this restricted domain. In the following, we show that this replacement does not affect our result regardless of the domain assumption. Let @ be the class of all solutions satisfying Ef, FR and Pr2. Lemma 4. Let (c, s) E .?A’with c, 2 cl 2 ... 2 c,, be given. For all i = 1, . . . , n and for all F’ E &, if F” = E, then lim,_W F/(c’, s) = s/n.

Fl!(c’-‘,s)~s/n for all tz2 and Proof. First note that, since F”= E, cf=maxj,, for all i. Also note that, since all F”s satisfy FR, we have C:ZC~I ...Lc; for all t, and therefore, Ff(c’,s)2s/n and F,‘(c’,s)
for all i,j,

c: - cj’ 2 F/(c2, s) - Fj’(c2, s) or equivalently, cf - c/’ - FI(c2, s) L -F,‘(c’, s) Since ci = s/n,

inequality

for all i,j.

(4)

to j = n gives

(3) applied

ct - i 2 F;(c2, s) - F,‘(c2, s)

for all i,

which by Ef gives

cf- ;

L

F;(c’, s) -

s -‘cl

Fj’(c2, s) ,

j=l

1

or equivalently, n-l 2F,‘(c2, s) 5 Ts+cf-

n-1 c F;(c2,s). j=2

‘0 I am grateful enables

to Professors

me to obtain

Lemma

4.

HervC Moulin

and Marcus

Berliant

for suggesting

this axiom

which

Y. Chun

Together

with inequality

/ Egalitarian

(4) applied

surplus

sharing

to j = 2, . . . , n - 1, we have n-1

n-l n 2F3c2, s) I -s+c;+

c

{c:-+F~(c2,s)},

j=2

or equivalently, n-l c c;-((n-2)FI’(cQ.

n-1 2FI’(c2, s) I -s+(n-l)c;-

j=2

As noted

we

in the beginning,

since

have n-2 -s, n

n-1 nF,‘(c2, s) I -s+(n-l)c:or equivalently,

Since this inequality

Similarly,

holds for all i, we have

we have 1

C;I--S+pC n2

n-l n

3 ”

or equivalently,

This argument,

generalized

to all t, for t = 2, . . . , yields

.:+~,~jl+...+(~~~*~~+(~)i~‘,:j.

Therefore,

we have s

- 5 lim ci 5 lim -

n

,+Ca

1

1

[-Q, n2 l-(n-1)/n

s2. n

257

258

Y. Chun / Egalitarian surplus sharing

This implies

that,

for all j,

lim F{(c’, s) = t. I’00 Finally,

by FR, we have lim F;‘(c’,s) = ;“2 for all i. fP+rn

0

It can easily be checked the replacement of Pr by Pr2 affects neither Lemma 2 nor Theorem 1. However, Pr cannot be completely dropped as shown in the following example. Let N= {1,2} and F be defined by

for all i, j= 1,2, i#j and for all (c,s) E 95’. It can easily be checked that this solution satisfies Ef and FR, but neither Pr not Pr2. Now suppose that c=(30, lo), s-20, F’ = B and F2 = E. Then we have c’= (30,lO) for all t = 1,. . . and the conflict is never resolved. To take care of the case n > 2, we introduce n - 2 additional agents identical to agent 1, set s = 10n and define P by

&(C,S)z(n-l)Cj-~

Cj+i

j+l n, if j and for all (c, s) E 6%‘.A similar argument foralli,j=l,..., flict is never resolved.

5. Extension

shows that the con-

to deficit sharing problems

Finally, we will discuss how our results can be applied to deficit sharing problems. A deficit sharing problem is a pair (c, s) E 8: x ‘8++ such that cN 2s> 0, with the following intended interpretation: each agent i, for i EN, has invested ci in some joint project, with results in a revenues. Unfortunately, the revenue is not sufficient to cover all investments. How should s be divided among agents? Equivalently, we could discuss how c, - s should be divided among agents. Let @ be the class of deficit sharing problems. All the results obtained so far carry over to this class of problems without any difficulty. However, the egalitarian solution becomes less appealing, since it violates the following natural requirement. No Free Lunch (NFL).

For all (c,s) E 9 and for all i, Fj(c,s)s

ci.

259

Y. Chun / Egalitarian surplus sharing

NFL requires that an agent should not gain from his investment, since the revenue is not sufficient to cover all investments. Let @* be the class of solutions satisfying Ef, FR, Pr and NFL. In the following, we will discuss how we can accomodate this additional axiom of NFL. As noted earlier, the egalitarian solution does not belong to S’*. Therefore, we introduce its variant, called the levelling solution, E*, first studied for deficit sharing problems by Young (1987,1988), defined in the following way. Let (c,s) E 9 with c, 2 c22 ... L c, be given.

Step 1. If c,,>s/n,

then E,*(c,s) =s/n

for all i. Otherwise,

E,*(c,s) = c, and go to

the next step.

Step 2. If S-Cc, c,-12p

’

n-l

then

E,*(c,s) = Otherwise,

s-n-l

for all i=l,...,n-1.

E,*_,(c,s) =c,_] and go to the next step . . . .

Step n-l. If c~>+(s-_CJ=~ cj), then E,*(c,s)=+(s--ET=3 wise, E,*(c,s)=c~ and E,*(c,s)=sCy=, Cj.

cJ) for i-1,2.

Other-

large Note that, since c, SS, E:(c, s) I cl. If the claim of each agent is sufficiently compared with the surplus, then the levelling solution yields the equal share of the surplus. Otherwise, each agent’s share is bounded from above by his claim. Our procedure applied to this problem with the additional axiom of NFL provides a noncooperative justification for the levelling solution. Given (c,s) E ~3, let IC N be the set of agents i such that ET(c,s)=c;, and let J=N\Z. Also, let s*=ET(c,s). We note that J= 0 implies that c,,, = s, and therefore, all solutions satisfying Ef and NFL assign c, to agent i, for all i. As a result, since the desired shares of agents are feasible, there is no conflict. On the other hand, if I= 0, then E*(c, s) = E(c, s) and, therefore, the conclusion of Lemma 1 can be applied. From now on, we assume that J#0 and I-0. Lemma 5. Let (c, s) E g with c, 2 c2 2 ... 2 c, be given. For all i = 1, . . . , n and for all F’E@*, if F”= E*, then lim,,, F,‘(c,s) = E;*(c,s). Proof. First note that, and for all i. Also note all t, and therefore, by for all t and for all i,j

since F”=E*, c,!=maxJE, F/(c’-‘,s)>E,*(c,s) for all tr2 that, since all F”s satisfy FR, we have c:?ciz ... zc: for Ef, FR and NFL, F;(c’,s)zzs*. Also by NFL, Fji(c’,s)< cj and since F” = E *, ci=cj for all tz2 and for all jEZ.

Y.Chun

260

/ Egalitarian

surplus

sharing

If we show that for all i, ,F,‘(c’,s) + ET(c, s) as t --) 03, then it follows, by Ef, FR and NFL, that for all i,j, F/‘(c’, s) + E;“(c, s) as t + 03. Therefore, we will only show that for all i, F,‘(c’,s) + E:(c,s) as t + co. Pr applied to the desired shares at step 2 implies that for all i,j.

c,‘F/(c2,s) 5 c~Fli(c2,s) Since cy = cj for all j E Z, equality

(5) applied

up these

to j E I gives

for all i and for all j E I.

cj F[(c2, s) 5 c:Fj’(c2, s) By summing

(3

IZl inequalities,

we have

c c,Ff(c2, s) 5 j;I c:$(c2, JEJ

s),

which by Ef gives C cjFt’(c2, S) 5 cf p pjFJ F;‘(C2, ‘)]

9

jeI

or equivalently,

Together

with inequality

(5) applied

to j E J, j#

1, we have

(ci+~~~,),:(c~,s)+F;(c~,i)~~~,~,c~~~Ci.

Since CjEJ,j+l

cJ!?(lJ1

-l)s*,

we have

c:+(jJ+l)s*+

cj F,‘(c2,s)sc~,

c jEI

(

>

or equivalently, SC;

F;(c2, s) 5 C:+((Jl-l)S*+Cj,,

Since this inequality

t+1

c,

holds for all i, we have SC;

cf5 This argument,

cj'

C:+(lJl-l)S*+Cj,, generalized

cj’

to all t, for t = 2, . . . , yields

SCf I

C:+(JJl-l)S*+Cj,,

Cj'

Y. Chun

Now, as in the proof of Lemma

f(x) =

Lemma

261

sharing

f: ‘iI?+--t $I+ defined

the function

by

SX cj’

Since the remaining

1. we omit it.

Remark.

surplus

1, we introduce

X+(IJ+l)S*+Cj,,

Note that f(s*) =s*. Lemma

/ Egalitarian

part of the proof

is the same as that of

0

4 also carries

over to this case in a similar

way.

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